To find the derivative of the function \( f(x) = e^{\frac{2}{x^3}} \), we can use the chain rule.

The chain rule states that if you have a function of the form \( f(g(x)) \), then the derivative is given by \( f'(g(x)) \cdot g'(x) \).

Here, we have \( f(x) = e^{u} \) where \( u = \frac{2}{x^3} \).

First, let's find the derivative of \( u = \frac{2}{x^3} \) with respect to \( x \).

\[ u = \frac{2}{x^3} = 2x^{-3} \]

The derivative of \( u \) with respect to \( x \) is:

\[ u' = \frac{d}{dx}(2x^{-3}) = 2 \cdot (-3)x^{-4} = -6x^{-4} \]

Next, let's find the derivative of \( e^u \) with respect to \( u \):

\[ \frac{d}{du}(e^u) = e^u \]

Now, using the chain rule:

\[ \frac{d}{dx} \left( e^{\frac{2}{x^3}} \right) = e^{\frac{2}{x^3}} \cdot \frac{d}{dx} \left( \frac{2}{x^3} \right) \]

Substitute \( u = \frac{2}{x^3} \) and \( u' = -6x^{-4} \):

\[ \frac{d}{dx} \left( e^{\frac{2}{x^3}} \right) = e^{\frac{2}{x^3}} \cdot (-6x^{-4}) \]

Thus, the derivative of the function \( f(x) \) is:

\[ f'(x) = -6x^{-4} \cdot e^{\frac{2}{x^3}} \]

So, the derivative of \( f(x) = e^{\frac{2}{x^3}} \) is:

\[ f'(x) = -\frac{6}{x^4} \cdot e^{\frac{2}{x^3}} \]

Do you want more details or have any questions? Here are 8 related questions that can further your understanding:

1. How can we use the derivative to find the critical points of \( f(x) \)?
2. What does the second derivative of \( f(x) \) tell us about the concavity of the function?
3. How can we evaluate the limit of \( f(x) \) as \( x \) approaches infinity?
4. What is the behavior of the function \( f(x) \) as \( x \) approaches 0?
5. Can we find the inflection points of \( f(x) \) using the second derivative?
6. How can we approximate \( f(x) \) using a Taylor series expansion around a specific point?
7. What are the asymptotic properties of \( f(x) \) for large values of \( x \)?
8. How can we use the derivative to sketch the graph of \( f(x) \)?

**Tip:** When finding the derivative of complex functions, breaking the problem into smaller parts and using the chain rule step-by-step can simplify the process.

Learn how to find the derivative of the function f(x) = e^(2/x^3) using the chain rule and derivative rules. This detailed solution covers differentiation of exponential functions and applications of the chain rule in calculus.

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